THE TITIUS-BODE FORMULA

In 1772 an astronomer named Titius living in Wittenberg
wrote to a colleague named Bode who was in Berlin.
Titius had discovered a very interesting relationship between the
average distances of the planets from the Sun.

Starting with the series 0, 3, 6, 12, 24,.. etc and then adding four and
dividing by 10, the resultant series was very close to the actual
distances of the planets from the Sun when measured in astronomical units.
(An astronomical unit or AU is the mean distance of the Earth from the Sun.)

The table below compares the actual distances to three significant figures
with the distances given by the series formula:

Note that neither Uranus nor Neptune were known at the time. Uranus
was not discovered until 1781, and Neptune not until 1846.
However, there was an anomaly, with the formula predicting that
a planet should exist at 2.8 AU.

Bode adopted this formula with so much enthusiasm that it came to be known
as Bode's Law. It was not until a few decades ago that historical
research restored the originator to equal status, and the formula is
now referred to in most books as the Titius-Bode law or rule.

Bode urged that a search be made for a planet at the 2.8 AU distance.
The formula was given the status of a "law" when William Herschal
discovered Uranus in 1781, and this was reinforced in 1801 when
Giuseppe Piazzi of Sicily discovered the first asteroid Ceres, which
happened to have a mean solar distance of 2.8 AU.

Unfortunately, this happy situation did not last. The error between
the formula prediction for Neptune was quite large. And for Pluto
it is enormous. In fact, those who still hold out some hope that
the formula really expresses some underlying physics of solar system
formation must have been very happy at the recent International
Astronomical Union (IAU) decision to exclude Pluto from the family
of planets and relegate it to a minor body, or dwarf planet.

We can express the Titius-Bode rule by the following mathematical formula:

D(n) = ( 3 x 2n + 4 ) / 10 AU

where n = -infinity, 0, 1, 2, 3, 4, .....

We unfortunately have to use a value of negative infinity for Mercury because
the initial series of Titius is not a true geometric series because of the
zero in the initial place.

It is also possible to write the Titius series in terms of kilometres rather
than astronomical units. We start off with the series 0, 45, 90, 180, 360, 720, ...
and then add 60 to each term. This gives us a planetary distance in millions
of kilometres, as seen below:

Despite many years of investigation, no explanation has been found for
any underlying reason that planets should or do follow this formula, and
most scientists now regard the Titius-Bode formula as just an interesting
near coincidence with reality.